Article
Statistical inference for generalized OrnsteinUhlenbeck processes
In this paper, we consider the problem of statistical inference for generalized OrnsteinUhlenbeck processes of the type
\[
X_{t} = e^{\xi_{t}} \left( X_{0} + \int_{0}^{t} e^{\xi_{u}} d u \right),
\]
where \(\xi_s\) is a L{\'e}vy process. Our primal goal is to estimate the characteristics of the L\'evy process \(\xi\) from the lowfrequency observations of the process \(X\). We present a novel approach towards estimating the L{\'e}vy triplet of \(\xi,\) which is based on the Mellin transform technique. It is shown that the resulting estimates attain optimal minimax convergence rates. The suggested algorithms are illustrated by numerical simulations.
Given a Lévy process (Lt)t≥0 and an independent nondecreasing process (time change) (T(t))t≥0, we consider the problem of statistical inference on T based on lowfrequency observations of the timechanged Lévy process LT(t). Our approach is based on the genuine use of Mellin and Laplace transforms. We propose a consistent estimator for the density of the increments of T in a stationary regime, derive its convergence rates and prove the optimality of the rates. It turns out that the convergence rates heavily depend on the decay of the Mellin transform of T. Finally, the performance of the estimator is analysed via a Monte Carlo simulation study.
In this paper, we introduce a principally new method for modelling the dependence structure between two L{\'e}vy processes. The proposed method is based on some special properties of the timechanged Levy processes and can be viewed as an reasonable alternative to the copula approach.
Given a Brownian motion B, we consider the socalled statistical Skorohod embedding problem of recovering the distribution of an independent random time T based on i.i.d. sample from BT. We propose a consistent estimator for the density of T, derive its convergence rates and prove their optimality.

Let a be a finite signed measure on [r,0], Z a Lévy process (that is a real process with independent stationary increments and càdlàg paths). A linear stochastic delay differential equation
X(t)=X(0)+∫ 0 t ∫ [r,0] X(s+u)da(u)ds+Z(t),t≥0,(1)driven by Z is studied, only càdlàg solutions to (1) such that Z and (X(t),r≤t≤0) are independent being considered. Set h(λ)=λ∫ [r,0] exp(λu)da(u) and v 0 =sup{Reλ∣λ∈ℂ,h(λ)=0}. Let the Lévy measure of jumps of the process Z be denoted by F. It is shown that there exists a stationary solution to (1) if and only if v 0 <0 and ∫ y>1 logydF(y)<∞. If X is a stationary solution to (1), then X(t) equals in law to ∫ 0 ∞ x 0 (t)dZ(t), where x 0 is the fundamental solution of the deterministic counterpart (Z≡0) to (1).
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible crosssection of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a crosssection exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a crosssection in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational crosssection in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational Wequivariant map T   >G/T where T is a maximal torus of G and W the Weyl group.